The “Brownian bees” model describes a system of $N$-independent branching Brownian particles. At each branching event the particle farthest from the origin is removed so that the number of particles remains constant at all times. Berestycki et al. [arXiv:2006.06486] proved that at $N\to \infty$ the coarse-grained spatial density of this particle system lives in a spherically symmetric domain and is described by the solution of a free boundary problem for a deterministic reaction-diffusion equation. Furthermore, they showed [arXiv:2005.09384] that, at long times, this solution approaches a unique spherically symmetric steady state with compact support: a sphere whose radius ${\ell }_0$ depends on the spatial dimension $d$. Here we study fluctuations in this system in the limit of large $N$ due to the stochastic character of the branching Brownian motion, and we focus on persistent fluctuations of the swarm size. We evaluate the probability density $\mathcal{P}(\ell ,N,T)$ that the maximum distance of a particle from the origin remains smaller than a specified value $\ell <{\ell }_0$ or larger than a specified value $\ell >{\ell }_0$ on a time interval $0<t<T$, where $T$ is very large. We argue that $\mathcal{P}(\ell ,N,T)$ exhibits the large-deviation form $-ln\mathcal{P}\simeq NT{R}_d\left(\ell \right)$. For all $d$'s we obtain asymptotics of the rate function $R_d\left(\ell \right)$ in the regimes $\ell \ll {\ell }_0,\ell \gg {\ell }_0$, and $|\ell -{\ell }_0|\ll {\ell }_0$. For $d=1$ the whole rate function can be calculated analytically. We obtain these results by determining the optimal (most probable) density profile of the swarm, conditioned on the specified $\ell$ and by arguing that this density profile is spherically symmetric with its center at the origin.

• Received 23 December 2020
• Accepted 5 March 2021

DOI:https://doi.org/10.1103/PhysRevE.103.032140